The purpose of this project is to conduct research in mathematical statistics, probability, and applied mathematics in order to develop new statistical methodology applicable to biomedical sciences. We are investigating the properties of different methods for displaying regional variability in cancer maps. We continue research into methods for designing epidemiologic studies with maximum power, deriving sample size formulas for testing homogeneity of relative risks in prospective studies. We also derived formulas for determining power and sample size requirements for equivalence testing based on the rate ratio estimated from matched samples. We are investigating methods of assessing inter-rater agreement, and have derived an efficient confidence interval estimate for the kappa coefficient and have established sample size requirements to attain sufficiently high power in reliability studies. We are continuing to investigate optimal methods for estimating the attributable risk, or etiologic fraction, and for calculating confidence intervals which correct for the negative bias in most current methods. A simulation study investigating the performance of a newly derived bias- corrected estimator of attributable risk continues. We also continue investigations into methods for examining birth cohort and calendar period trends in disease rates, continuing a simulation study of the performance of two novel, exact permutation tests for changes in the slope of the birth cohort trend. A computer program to implement the exact non- parametric methods for examining birth cohort trends is being improved, and development continues on a computer program implementing parametric methods for examining birth cohort and calendar period patterns of risk in age-period-cohort models. We continue developing methods for examining mutational spectra in a defined DNA sequence, including tests for whether a certain pattern of mutations occurs more frequently than expected by chance in a specific gene.